Finite-Part Integrals and Modified Splines

We first state a uniform convergence theorem for finite-part integrals which are derivatives of weighted Cauchy principal value integrals. We then give a two-stage process to modify approximating splines and optimal nodal splines in such a way that the conditions of this theorem are satisfied. Consequently, these modified splines can be used in the numerical evaluation of these finite-part integrals.     

样条共轭插值与L_p模误差估计

[1—5]讨论了各种类型插值样条的L_∞模最优误差估计。本文利用共轭插值样条,给出一些插值样条类的L_1模最优误差界,然后用插值空间理论导出L_p模估计的上界。 一、样条共轭插值 设n≥1并给定[0,1]上的两个分划:     

High order convergence for collocation of second kind boundary integral equations on polygons

We propose collocation methods with smoothest splines to solve the integral equation of the second kind on a plane polygon. They are based on the bijectivity of the double layer potential between spaces of Sobolev type with arbitrary high regularity and involving the singular functions generated by the corners. If splines of order are used, we get quasi-optimal estimates in -norm and optimal order convergence for the -norm if . Numerical experiments are presented. Received November 20, 1996 / Accepted March 10, 1997     

Some extremal properties of multivariate polynomial splines in the metric Lp (Rd )

We constructed a kind of continuous multivariate spline operators as the approximation tools of the multivariate functions on the Bd instead of the usual multivariate cardinal interpolation oper-ators of splines, and obtained the approximation error by this kind of spline operators. Meantime, by the results, we also obtained that the spaces of multivariate polynomial splines are weakly asymptoti-cally optimal for the Kolmogorov widths and the linear widths of some anisotropic Sobolev classes of smooth functions on Bd in the metric Lp(Bd).     

Optimal smoothing and interpolating splines with constraints

This paper considers the problem for designing optimal smoothing and interpolating splines with equality and/or inequality constraints. The splines are constituted by employing normalized uniform B-splines as the basis functions, namely as weighted sum of shifted B-splines of degree k. Then a central issue is to determine an optimal vector of the so-called control points. By employing such an approach, it is shown that various types of constraints are formulated as linear function of the control points, and the problems reduce to quadratic programming problems. We demonstrate the effectiveness and usefulness by numerical examples including approximation of probability density functions, approximation of discontinuous functions, and trajectory planning.     

To the theory of optimal splines

On June 18, 2008 at the Plenary Meeting of the International Conference “Differential Equations and Topology” dedicated to the 100th anniversary of Pontryagin, the report [1] was submitted by Isaev and Leitmann. This report in a summary form included a section dedicated to the research of scientists of TsAGI in the field of automation of full life-cycle (i.e. engineering-design-manufacturing, or CAE/CAD/CAM, or CALS-technologies) of wind tunnel models [2]. Within this framework, methods of geometric modeling [3] and [4] were intensively developed, new classes of optimal splines have been built, including the Pontryagin splines and the Chebyshev splines [5], [6], [7] and [8]. This paper reviews some results on the Pontryagin splines. We also give some results on the Lurie splines, that arise in the problem of interpolation of a cylindrical type surface given by the family of table coplanar planes.     

带障碍的广义插值样条与带状态约束的最优控制

本文由样条的极值性质出发给同分算子插值样条（即广义插值样条）新的推导方法。用这种方法可推导出带障碍（即带不等式约束）的微分算子插值样条的解析性质,为简便计,本文以非负广义插值样条为例。最后,揭示了状态带不等式的最优控制解的必要性准则与带障碍的广义插值样条的联系。     

Free-knot spline approximation of stochastic processes

We study optimal approximation of stochastic processes by polynomial splines with free knots. The number of free knots is either a priori fixed or may depend on the particular trajectory. For the s-fold integrated Wiener process as well as for scalar diffusion processes we determine the asymptotic behavior of the average L_p-distance to the splines spaces, as the (expected) number of free knots tends to infinity.     

The derivation of cubic splines with obstacles by methods of optimization and optimal control

Summary Interpolating splines which are restricted in their movement by the presence of obstacles are investigated. For simplicity we mainly treat cubic splines which are required to be non-negative. The extension to splines of higher order and to certain other forms of obstacles is straightforward. Methods of optimization and of optimal control are used to obtain necessary optimality criteria. These criteria are applied to derive an algorithm to compute splines which are restricted to constant lower or upper bounds. There is a numerical example which illustrates the method presented.Dedicated to Günter Meinardus on the occasion of his 60th birthday     

On the structure of function spaces in optimal recovery of point functionals for ENO-schemes by radial basis functions

Summary. Radial basis functions are used in the recovery step of finite volume methods for the numerical solution of conservation laws. Being conditionally positive definite such functions generate optimal recovery splines in the sense of Micchelli and Rivlin in associated native spaces. We analyse the solvability to the recovery problem of point functionals from cell average values with radial basis functions. Furthermore, we characterise the corresponding native function spaces and provide error estimates of the recovery scheme. Finally, we explicitly list the native spaces to a selection of radial basis functions, thin plate splines included, before we provide some numerical examples of our method. Received March 14, 1995     

Asymptotic analysis of boundary conditions for quintic splines

In this article, we consider various boundary conditions for interpolation of quintic splines of defect 1 on a uniform mesh. We obtain an asymptotic representation of the approximation error for the spline for different boundary conditions. Boundary conditions that are optimal by approximation accuracy are found.     

Local Lagrange Interpolation with Bivariate Splines of Arbitrary Smoothness

We describe a method which can be used to interpolate function values at a set of scattered points in a planar domain using bivariate polynomial splines of any prescribed smoothness. The method starts with an arbitrary given triangulation of the data points, and involves refining some of the triangles with Clough-Tocher splits. The construction of the interpolating splines requires some additional function values at selected points in the domain, but no derivatives are needed at any point. Given n data points and a corresponding initial triangulation, the interpolating spline can be computed in just O(n) operations. The interpolation method is local and stable, and provides optimal order approximation of smooth functions.     

Mathematical Modeling of Human Walking on the Basis of Optimization of Controlled Processes in Biodynamical Systems

We propose a new statement of the problem of optimal control over a nonlinear dynamical system with phase restrictions concerning the mathematical modeling of human walking. We develop an algorithm for obtaining an approximate solution of the formulated problem of optimal control on the basis of the parametrization of independently varied functions by cubic smoothing splines and on the basis of the minimization of the objective function in orthogonal directions. The efficiency of the algorithm is illustrated by the numerical simulation of human walking at a normal pace along a horizontal surface.     

Some extremal properties of multivariate polynomial splines in the metric Lp(ℝd)

We constructed a kind of continuous multivariate spline operators as the approximation tools of the multivariate functions on the (?^d instead of the usual multivariate cardinal interpolation operators of splines, and obtained the approximation error by this kind of spline operators. Meantime, by the results, we also obtained that the spaces of multivariate polynomial splines are weakly asyrnptotically optimal for the Kolrnogorov widths and the linear widths of some anisotropic Sobolev classes of smooth functions on (?^d in the metric L_p((?^d).     

几种有理插值函数的逼近性质

1　引　　言在曲线和曲面设计中,样条插值是有用的和强有力的工具.不少作者已经研究了很多种类型的样条插值[1,2,3,4].近些年来,有理插值样条,特别是三次有理插值样条,以及它们在外型控制中的应用,已有了不少工作[5,6,7].有理插值样条的表达式中有某些参数,正是由于这些参数,有理插值样条在外型控制中充分显示了它的灵活性;但也正是由于这些参数,使它的逼近性质的研究增加了困难.因此,关于有理插值样条的逼近性质的研究很少见诸文献.本文在第二节首先叙述几种典型的有理插值样条,其中包括分母为一次、二次的三次有理插值样条和仅基于函数值…     

A Family of Spline Quasi-Interpolants on the Sphere

In this paper, we construct a local quasi-interpolant Q for fitting a function f defined on the sphere S. We first map the surface S onto a rectangular domain and next, by using the tensor product of polynomial splines and 2-periodic trigonometric splines, we give the expression of Qf. The use of trigonometric splines is necessary to enforce some boundary conditions which are useful to ensure the C ² continuity of the associated surface. Finally, we prove that Q realizes an accuracy of optimal order.     

Construction of splines of maximal smoothness

We consider approximation relations as a system of linear algebraic equations and obtain spaces of minimal Lagrange type splines of arbitrary order. Using an analog of procedure for constructing elementary symmetric polynomials, we obtain new explicit formulas for representing splines. The splines obtained possess the maximal smoothness and minimal compact support. We also give examples of constructing splines on an open interval and on a segment. Bibliography: 15 titles.     

Lagrange Interpolation by Bivariate C 1-Splines with Optimal Approximation Order

We develop the first local Lagrange interpolation scheme for C ¹-splines of degree q≥3 on arbitrary triangulations. For doing this, we use a fast coloring algorithm to subdivide about half of the triangles by a Clough–Tocher split in an appropriate way. Based on this coloring, we choose interpolation points such that the corresponding fundamental splines have local support. The interpolating splines yield optimal approximation order and can be computed with linear complexity. Numerical examples with a large number of interpolation points show that our method works efficiently.     

A stochastic framework for recursive computation of spline functions: Part II,smoothing splines

Many of the optimal curve-fitting problems arising in approximation theory have the same structure as certain estimation problems involving random processes. We develop this structural correspondence for the problem of smoothing inaccurate data with splines and show that the smoothing spline is a sample function of a certain linear least-squares estimate. Estimation techniques are then used to derive a recursive algorithm for spline smoothing.     

Optimal Interpolation of Scattered Data on a Circular Domain with Boundary Conditions

Optimal interpolation problems of scattered data on a circular domain with two different types of boundary value conditions are studied in this paper. Closed-form optimal solutions, a new type of spline functions defined by partial differential operators, are obtained. This type of new splines is a generalization of the well-known $L_g$-splines and thin-plate splines. The standard reproducing kernel structure of the optimal solutions is demonstrated. The new idea and technique developed in this paper are finally generalized to solve the same interpolation problems involving a more general class of partial differential operators on a general region.